Lotto Probability

[BPScooter]

OK everybody, I'm now sort of confused. We've gone from discussing a lotto-style probability game (which can be better discussed by mathemeticians than me) to a heads/tails prediction game (which does have intuitive appeal because it boils down to probability) to a chaos sort of problem (the billiard balls) and probably others.

Are we interested in the patterns themselves, or the way that people interpret the patterns?

Yes, yes, I understand that the only way I can perceive a pattern is by being a rational being, and that some of these patterns are not "real" in the sense that I may be convinced of them, yet they do not lend themselves to empirical proofs. I think it was Locke that tried to nail that down a couple hundred years ago. Let me know if you have updates on that program. I think he got it pretty well right. Probably further conversation in that regard needs to be on the "Religion and Philosophy" board.

 

[BillyJoe]

...a DNA combination of a particular individual is rare -- as they are for everyone. But they (DNA codes) are not improbable, for they happen all the time -- each time someone is conceived a very rare combination occurs. But that occurance (of a rare DNA code) is not an improbable event. Expecting that code to form beforehand and predicting it exactly would be the improbable event -- not the formation of it. That doesn't happen all that often, if ever. But rare DNA combinations occur quite often.

Well, as far as I'm concerned, there's no way round it: An improbable event is rare. A rare event is improbable. How can that not be so? Consider this quote from your post....

"a DNA combination of a particular individual is rare.....But they (DNA codes) are not improbable"

You are mixing singles with pleurals.
A (particular) DNA combination is improbable and rare.
DNA codes (as a group) are not improbable and not rare.
Mix up the singles and pleurals and you can get rare but not improbable.
Neat trick.

But perhaps it's just me. <shrug shoulders>
For example, I just do not understand this...

"that occurance (of a rare DNA code) is not an improbable event"
"rare DNA combinations occur quite often"

As I say, perhaps it;s just me.

BillyJoe

 

[Just thinking]

... But perhaps it's just me. <shrug shoulders>
For example, I just do not understand this...

"that occurance (of a rare DNA code) is not an improbable event"
"rare DNA combinations occur quite often"

As I say, perhaps it;s just me.

BillyJoe

Try substituting unique for rare and perhaps you'll see my point better.

The production of unique humans is not improbable -- they happen all the time. It's once you try to describe a particular human to be produced, then it is not just unique -- it's improbable. I'm not really mixing singles with plurals, I'm describing having a unique event occur as compared to having it predicted beforehand.

If we take the old proverb of all snowfalkes are different, then all snowflakes are unique and each is rare. But they are produced in abundance -- even making one at a time is no big deal. So making a snowflake is not improbable -- even making a unique (rare) one, which they all are. But as soon as you say "I want to see this particular pattern of snowflake", then we see how making a unique snowflake is now highly improbable.

 

[Rasmus]

For example, I just do not understand this...

"that occurance (of a rare DNA code) is not an improbable event"

"that occurance (of aany one of a gazillion possible rare DNA code) is not an improbable event"

i.e. it is very likely that this DNA sequence will be human rather than frog, if the parents are also human.

"rare DNA combinations occur quite often"

Every single given DNA combination is in itself rare, most are unique. Still, it happens a lot that people have a unique DNA combination. In fact, it happens with mostly everybody.

Or, going back to the lottery: Week after week after week, they draw an extremely unlikely comnbination of winning balls.

 

[Just thinking]

... Or, going back to the lottery: Week after week after week, they draw an extremely unlikely comnbination of winning balls.

Ahhh ... I do believe you and I are on the same page with this in trying our best to get BillyJoe on board, but I think your choice of words with "extremely unlikely" implies it to be improbable (the balls drawn), not just very rare. It is very likely (heck, it's almost certain) that they will draw a combination of winning balls (they do it every week, or whatever); what they frequently draw is a rare or unique combination of winning balls.

What is "extremely unlikely" is having a ticket with numbers matching those on the winning balls, or having those exact same balls come up over and over week after week. You see, in both examples we needed another event to compare the drawing to, which then can be described as "extremely unlikely". But the event taken to only itself (the drawing of winning lotto balls) is not unlikely -- even though that specific outcome of numbers will be rare.

Another way to consider the drawn combination to not be "extremely unlikely" is to realize that all combinations share the same probability of occuring, hence no one combination is less likely than any other.

Let's face it -- this can be hard to describe.

 

[Rasmus]

Another way to consider the drawn combination to not be "extremely unlikely" is to realize that all combinations share the same probability of occuring, hence no one combination is less likely than any other.

Let's face it -- this can be hard to describe.

I still don't fully agree with you.

It might be semantics, but any one drawn combination is "extremely unlikely" in itself. I am still not certain what point you are trying to make here - and it looks to me that it still might be that a combination of many many unlikely events is less unlikely to occur than just one unlikely event.

 

[BillyJoe]

Well, hell, I think we are ALL on the same page. It's just that we are disagreeing with the way we are describing what we agree on. (The casualty has been mummypyjamas who has probably gone on to study synchronicity by now).

My original statement was...

"Improbable things happen all the time."

But you must remember I was repying to mummypyjamas who had two improbable events occurring together and wondered at the astounding odds of this happening. I replied that "improbable events (like winning lotto or lightning striking, or a particular hand in bridge, or a particular distribution of billiard balls, or a particular snowflake pattern) happen all the time".
(I was going to go on to explain - if he expressed any interest - that two improbable events, such as these, occurring together are just a coincidence which is bound to happen eventually considering that improbable events happen all the time.)

Perhaps it would have been clearer if I had stated:

Things that are members of the class "Improbable things" happen all the time.

BJ

 

[Just thinking]

Actually, at one time I was purplexed myself at what happened each and every time I made a break on a billiard table. (I used to have one at my parent's house, so I played often.) After the break, I would say, "Wow -- look at that. The chances of everything occuring just like that were billions to 1 -- yet it just happened. And I could do that over and over again. How could it be that I can do these improbable things so easily?"

But now I realize that I was describing the situation incorrectly. Each result from a break was surely very rare -- if not unique -- but making a result (not a specific one -- just one in general) was not improbable, for a distribution of balls was going to happen each and every time. Nor was making a distribution "extremely unlikely". However, expecting a specific distribution to occur before I broke the balls apart is what would be highly improbable, or "extremely unlikely". This is what is being said by Rasmus with "any one drawn combination is "extremely unlikely" in itself". It is implying a specific distribution -- predicted in an indirect way. He is emphasizing one outcome -- which will be highly unlikely -- not just an outcome in general. This is where the problem of semantics (in my opinion) lies.

 

[tsg]

But now I realize that I was describing the situation incorrectly. Each result from a break was surely very rare -- if not unique -- but making a result (not a specific one -- just one in general) was not improbable, for a distribution of balls was going to happen each and every time.

This is where the problem lies: Looking at a specific event and asking, "what were the odds of that happening?" as though it were something amazing discounts the thousands, or millions, of other events that would also qualify as "something amazing". "Extremely improbable events" do happen all the time because they have lots and lots of chances to. There are also lots and lots of extremely improbable events to choose from. When you take into account the number of times improbable events have to occur, as well as the number of events that would qualify as "amazing", the probability of "something amazing" happening becomes pretty likely indeed.

 

[Just thinking]

This is where the problem lies: Looking at a specific event and asking, "what were the odds of that happening?" as though it were something amazing discounts the thousands, or millions, of other events that would also qualify as "something amazing".

Only if any one of them was specifically predicted to happen beforehand -- not if just any one of them is the one we see unfold in front of us.

"Extremely improbable events" do happen all the time because they have lots and lots of chances to. There are also lots and lots of extremely improbable events to choose from. When you take into account the number of times improbable events have to occur, as well as the number of events that would qualify as "amazing", the probability of "something amazing" happening becomes pretty likely indeed.

I think you are falling into the same trap as some others have. Extremely improbable events do not have lots and lots of chances to happen -- but extremely rare events (taken as a group) do, because there are so many of them. The "amazing" quality of a rare event (or being of a highly improbable nature) is only attached to the rare event if it is tied somehow to an unusual frequency of occurance. Seeing a set of 6 random lotto balls come out as a winning combination is not amazing -- but if those exact same 6 came out again the very next week, then it is amazing. If even you win the big prize, it's not amazing (as many times someone surely does). But win it week after week and then you've got something. Or tell us in advance which specific lotto you will win, and then it's amazing -- but just playing and eventually winning won't do it.

Just being rare does not automatically make something improbable -- but something improbable does imply a good degree of rarity. However, seeing the exact same rare event repeat with very few trials is improbable. What's "amazing" about something rare is when it repeats often -- not just once.

 

[BillyJoe]

Hey, JT, we AGREE with you!
When tsg writes, in his last post...

"Extremely improbable events" do happen all the time because they have lots and lots of chances to. There are also lots and lots of extremely improbable events to choose from.

He means the same as when you write...

....extremely rare events (taken as a group) do [have lots and lots of chances to happen ], because there are so many of them.

He means "extremely improbable events - as a group"!
He AGREES with you!
And so was I when, in my last post, I wrote:
"Events that are members of the class "Improbable events" happen all the time."
This sounds a lot like your version:
"extremely rare events (taken as a group) "
Don't you think? (Compare the highlighted bits.)
See - we AGREE!


However, we do "DISAGREE" here.....

Seeing a set of 6 random lotto balls come out as a winning combination is not amazing -- but if those exact same 6 came out again the very next week, then it is amazing.

What if the "6 random lotto balls" that came out happened to be 1,2,3,4,5,6? Would you still need to wait till the next draw and see this outcome repeated before you would be amazed? (Or, if six consecutive numbers in lotto are not amazing enough for you, what if that lousy bidge hand you were dealt last week actually came out as 13 hearts? Would you be amazed? Or would you need to see it repeated in the next hand before you would be amazed?)

So, why are we amazed with a single appearance of 1,2,3,4,5,6 (or thirteen hearts) but not amazed with an odd selection of numbers unless it is repeated next week? They have exactly the same odds after all. The answer is that they have a recognisable pattern, whilst an odd selection of numbers looks like any other odd selection of numbers. But they are not. Each set of six numbers have the same odds of 1 in 13 million. It is only a coincidence (as I said to mummypyjamas) that this particular set (1,2,3,4,5,6) came up. If the balls were coloured, instead of numbered, nobody would be amazed at any outcome because there would not be any recognisable patterns.

This is what I meant when I said to mummypyjamas that improbable events (OR "events that are members of the group 'improbable events'", OR "improbable events as a group") happen all the time and that the pattern he observed was just coincidence.


BillyJoe

 

[Just thinking]

Hey, JT, we AGREE with you!
When tsg writes, in his last post...

He means the same as when you write...

He means "extremely improbable events - as a group"!
He AGREES with you!

There is no doubt about it, we are very close on this. But in my heart I still have this unsure feeling that the two of you may occasionally treat the concepts rare and improbable as interchangable -- they are not. According to the ]Merriam-Webster Online Dictionary ...

Improbable : unlikely to be true or to occur

Rare : seldom occurring or found

We see that it mentions nothing unlikely to occur about being rare. This is exactly what I have been claiming. Each rare quantity will be seldom in event -- but because there are many possible outcomes in random events, one of those will occur, and we will consider its occurance rare due to all the other possible outcomes that may also have occured. But with equal likleyhood -- and that's the rub. Because all the outcomes have equal probabilities of occuring, the one we saw was not any more or less improbable than all the others. But it was just as rare as any other.

And so was I when, in my last post, I wrote:
"Events that are members of the class "Improbable events" happen all the time."
This sounds a lot like your version:
"extremely rare events (taken as a group) "
Don't you think? (Compare the highlighted bits.)

By now I must sound like an old stubborn _____ (fill in the blank)!

Yes, it does sound a lot like what I am saying, but there are differences. The class of Improbable Events is (as described) a class of possible outcomes -- they only become individually improbable when you attach some other condition upon a specific one ... it must match my lotto ticket * it must occur twice in a row * it must have a specific pattern that I recognize * etc. By just turning up with no specific requirement is no more improbable than picking out a jellybean from a jar of thousands. You can't say that (after picking) picking that specific one was thousands to one against and I just did a very improbable thing because all you did was paint the bullseye around your shot. It ws not improbable to pick a jellybean -- it would be improbable to somehow indicate your picking a specific one beforehand, and then doing it.

See - we AGREE!

However, we do "DISAGREE" here.....

What if the "6 random lotto balls" that came out happened to be 1,2,3,4,5,6? Would you still need to wait till the next draw and see this outcome repeated before you would be amazed? (Or, if six consecutive numbers in lotto are not amazing enough for you, what if that lousy bidge hand you were dealt last week actually came out as 13 hearts? Would you be amazed? Or would you need to see it repeated in the next hand before you would be amazed?)

Actually that's a good question (for me to address). If the numbers 1,2,3,4,5,6 came up (or that bridge hand) I would not be as amazed as you might think. Why? Because each of those events has no less a chance of happening as any other 6 numbers or bridge hand. Agree? If you do, and I think you do, then you must see that they are really not amazing. Yes, I would be amazed if they repeated with few trials, but you would be (should be) just as amazed if it turned out for any other 6 numbers or bridge hand to repeat exactly as it did just prior. But keep in mind, it is improbable to get a pattern as compared to all the other outcomes -- but if one does occasionally turn up, that is not so amazing. It is not improbable for improbable events to occasionally happen. (Please re-read that.)

So, why are we amazed with a single appearance of 1,2,3,4,5,6 (or thirteen hearts) but not amazed with an odd selection of numbers unless it is repeated next week? They have exactly the same odds after all. The answer is that they have a recognisable pattern, whilst an odd selection of numbers looks like any other odd selection of numbers. But they are not. Each set of six numbers have the same odds of 1 in 13 million. It is only a coincidence (as I said to mummypyjamas) that this particular set (1,2,3,4,5,6) came up. If the balls were coloured, instead of numbered, nobody would be amazed at any outcome because there would not be any recognisable patterns.

Here we fully agree.

This is what I meant when I said to mummypyjamas that improbable events (OR "events that are members of the group 'improbable events'", OR "improbable events as a group") happen all the time and that the pattern he observed was just coincidence.

I guess I just don't like using improbable with rare so interchangably.

 

[Rasmus]

Yes, it does sound a lot like what I am saying, but there are differences. The class of Improbable Events is (as described) a class of possible outcomes -- they only become individually improbable when you attach some other condition upon a specific one ... it must match my lotto ticket * it must occur twice in a row * it must have a specific pattern that I recognize * etc. By just turning up with no specific requirement is no more improbable than picking out a jellybean from a jar of thousands. You can't say that (after picking) picking that specific one was thousands to one against and I just did a very improbable thing because all you did was paint the bullseye around your shot. It ws not improbable to pick a jellybean --

But it was improbable that that particular jellybean was picked. It doesn't make a difference, whether anyone had pointed at it beforehand, or tried to predict the jellybean.

it would be improbable to somehow indicate your picking a specific one beforehand, and then doing it.

That would also be improbable, but it describes a totally different event.

We are looking at three different things, really:

- choosing any one jellybean.
It is neither improbably not rare.

- choosing a particular jellybean
It is both improbable and rare, from what I can tell. With a thousand jellybeans in the jar, chances are 1/1000 and the jellybean will be picked once every 1000 times.

- predicting which jellaybean will be chosen.
same as above: It happens once every 1000 times and hence is improbably and rare.

That has nothing to do with drawing bullseyes after the dart was thrown. though. I think you are mixing up "remarkable" or "special" with "improbable". What we perceive to be special is entirely subjective; a lot of people will agree that the lottery numbers 1-2-3-4-5-6-7 are special; 2-3-5-6-10-49 will bear little significance for anyone. (They would be significant for me; they are - in increasing order - made up from the coordinates of a former address of mine)

Rasmus.

 

[Just thinking]

But it was improbable that that particular jellybean was picked. It doesn't make a difference, whether anyone had pointed at it beforehand, or tried to predict the jellybean.

Now we have a leg to stand on.

It does make a difference -- look what you say ... 'But it was improbable that that particular jellybean was picked." You have given that jellybean significance in singularizing it. You are painting a bullseye around your shot. It's like shooting at the side of a blank wall and then going up to the point of entry and saying ... "Wow -- what are the chances of my hitting the wall at that spot!" Yes, it was improbable if you consider that location special in some way (a bullseye already there; your predicting the spot beforehand; the exact same position as a previous random shot; etc.). But just hitting the wall, picking a jellybean, getting a set of lotto numbers, drawing a bridge hand, etc. are not improbable events -- even given that each one is rare compared to all the others.

That would also be improbable, but it describes a totally different event.

I disagree -- once you say "that particular" event, you have changed the problem entirely.

We are looking at three different things, really:

- choosing any one jellybean.
It is neither improbably not rare.

I Agree.

- choosing a particular jellybean
It is both improbable and rare, from what I can tell. With a thousand jellybeans in the jar, chances are 1/1000 and the jellybean will be picked once every 1000 times.

I Agree.

- predicting which jellaybean will be chosen.
same as above: It happens once every 1000 times and hence is improbably and rare.

I Agree -- but this is exactly what you have described in the previous case. Once you state a "particular" event, (or an event as being "particular") you have the same thing as a prediction -- you have given that event significance of some kind -- singling it out. Otherwise, you are back to your first case. Consider the detective looking for clues at a crime scene ... he discovers something rare -- a casino chip from some other country. He goes to a suspect's house and searches it finding another chip just like the one at the crime scene. This chip (rare as it is) would have no significance in this person's house had it not been at the crime scene -- but becasue we now have two rare events matching, our detective considers the events as being highly improbable in there being no connection.

That has nothing to do with drawing bullseyes after the dart was thrown. though. I think you are mixing up "remarkable" or "special" with "improbable". What we perceive to be special is entirely subjective; a lot of people will agree that the lottery numbers 1-2-3-4-5-6-7 are special; 2-3-5-6-10-49 will bear little significance for anyone. (They would be significant for me; they are - in increasing order - made up from the coordinates of a former address of mine).

I consider remarkable and special as rare, but not always improbable; some may be and some may not be -- it all depends on how the event is described. Even if your former address numbers won you the lottery, that would not be amazing -- improbable, yes. (For you -- if someone else just played those numbers and won there would be less amazement. Why? Because there are far greater numbers of people playing that have no significance of those numbers as compared to the one you that does). Win with those two weeks in a row and it's now amazing.

 

[BillyJoe]

....in my heart I still have this unsure feeling that the two of you may occasionally treat the concepts rare and improbable as interchangable -- they are not. According to the ]Merriam-Webster Online Dictionary ...

Improbable : unlikely to be true or to occur

Rare : seldom occurring or found.

Improbable: unlikely to occur
Rare: seldom occurring.
Improbable is before the event. Rare is after the event.
An improbable event (meaning an event that is unlikely to occur) seldom occurs (meaning that it is rare).

The class of Improbable Events is (as described) a class of possible outcomes -- they only become individually improbable when you attach some other condition upon a specific one ... it must match my lotto ticket * it must occur twice in a row * it must have a specific pattern that I recognize * etc. .

Individually, a set of six lotto numbers has odds of 1 in 13 million. That is improbable! - regardless of whether it matches your lotto ticket, or has a pattern, or occurs twice in a row. On it's own it has the improbable odds of 1 in 13 million!

. By just turning up with no specific requirement is no more improbable than picking out a jellybean from a jar of thousands. You can't say that (after picking) picking that specific one was thousands to one against and I just did a very improbable thing because all you did was paint the bullseye around your shot. It ws not improbable to pick a jellybean -- it would be improbable to somehow indicate your picking a specific one beforehand, and then doing it. .

I think you're mixing up before and after.
Before the lotto draw each set of six lotto numbers has the improbable odds of 1 in 13 million. After the draw probability doesn't apply - it has already happened. However it is ligitimate to say that the set of six lotto numbers will be found rarely - you'll generally have to wait a long time to see them come out again.
That is: An improbable event happens rarely.

Actually that's a good question (for me to address). If the numbers 1,2,3,4,5,6 came up (or that bridge hand) I would not be as amazed as you might think.

You would be 1 in 6 billion!

Why? Because each of those events has no less a chance of happening as any other 6 numbers or bridge hand. Agree?.

I agree that "each of those events has no less a chance of happening as any other 6 numbers or bridge hand", but not that this is a reason not to be amazed when 1,2,3,4,5,6 or thirteen hearts come up.

Yes, I would be amazed if they repeated with few trials...

It would take two bridge hands of thirteen hearts within a few deals to amaze you??? JT, you are destined never ever to be amazed. the odds are about 1 in 100,000,000,000,000,000,000!

But keep in mind, it is improbable to get a pattern as compared to all the other outcomes -- but if one does occasionally turn up, that is not so amazing.

The clue as to why it is amazing when 1,2,3,4,5,6 turns up in lotto rather than any odd set of numbers (or when 13 hearts are dealt instead of any odd collection of thirteen cards) is PATTERN, as Jaggy Bunnet correctly replied some time ago.
The reason is that outcomes that have a pattern are like predicted outcomes. A pattern (like 1,2,3,4,5,6 or thirteen hearts) is immediately recognisable. These patterns sit there in your brain. When 1,2,3,4,5,6 or thirteen hearts appear in real life, they match patterns sitting in your brain, almost as if you have predicted them. That's why you are amazed.


regards,
BillyJoe

 

[BillyJoe]

Cancelled post

 

[Just thinking]

Improbable: unlikely to occur
Rare: seldom occurring.
Improbable is before the event. Rare is after the event.
An improbable event (meaning an event that is unlikely to occur) seldom occurs (meaning that it is rare).

BJ, are you reading what I post? I have always contended that improbable events are rare, but not all rare events are improbable. Where does it say in the definition that rare events are unlikely, along with being seldom?

Individually, a set of six lotto numbers has odds of 1 in 13 million. That is improbable! - regardless of whether it matches your lotto ticket, or has a pattern, or occurs twice in a row. On it's own it has the improbable odds of 1 in 13 million!

No -- Drawing any set of six numbers is not improbable; drawing a specific set is. When you say "a set" you are specifying something special about that one set as compared to all the others. If not, then you are disagreeing with yourself.

I think you're mixing up before and after.
Before the lotto draw each set of six lotto numbers has the improbable odds of 1 in 13 million. After the draw probability doesn't apply - it has already happened. However it is ligitimate to say that the set of six lotto numbers will be found rarely - you'll generally have to wait a long time to see them come out again.
That is: An improbable event happens rarely.

It will be improbable for a given set of numbers to repeat often, and also rare, so as put this way -- we agree. But whether I single out a set of numbers before or after as being specific, it makes no difference as to its likelyhood of occuring (as a specific set); but for any set to occur at all is not improbable.

You would be 1 in 6 billion!

I doubt that -- as many others (but certainly not a majority) would know that those numbers have just as much chance of occuring as any other single combination. Is it really so amazing to you -- knowing that? Do you think it would get the $1,000,000 JREF prize?

I agree that "each of those events has no less a chance of happening as any other 6 numbers or bridge hand", but not that this is a reason not to be amazed when 1,2,3,4,5,6 or thirteen hearts come up.

Use colors instead of numbers and all amazement fades away. (Unless they match the ROY-G-BIV pattern, right? )

It would take two bridge hands of thirteen hearts within a few deals to amaze you??? JT, you are destined never ever to be amazed. the odds are about 1 in 100,000,000,000,000,000,000!

Did you read what I asked you to please read twice? Even Randi is in agreement with that, although I can't point to his exact quote. Besides, those odds are exactly the same for any other single hand! Amazed at those? Why not? You're painting a bullseye around the shot. Think about what happens -- you're delt the "amazing" hand. Your eyes look at it, then your brain sees it. Then your brain recognizes that all the cards have one thing in common. Then you realize that this has significance as being a perfect hand. All those things happened after the cards came to you with no special ability in doing so. You made all the significance to the hand after you got it -- you painted your bullseye! That exact same hand would have much less value or significance in playing UNO.

The clue as to why it is amazing when 1,2,3,4,5,6 turns up in lotto rather than any odd set of numbers (or when 13 hearts are dealt instead of any odd collection of thirteen cards) is PATTERN, as Jaggy Bunnet correctly replied some time ago.
The reason is that outcomes that have a pattern are like predicted outcomes. A pattern (like 1,2,3,4,5,6 or thirteen hearts) is immediately recognisable. These patterns sit there in your brain. When 1,2,3,4,5,6 or thirteen hearts appear in real life, they match patterns sitting in your brain, almost as if you have predicted them. That's why you are amazed.

Perhaps it's because of the above I am not amazed so much.

 

[BillyJoe]

BJ said: Individually, a set of six lotto numbers has odds of 1 in 13 million. That is improbable!

JT replied: No -- Drawing any set of six numbers is not improbable; drawing a specific set is.

You forgot to read the highlighted bit. When you do , you will see that I said exactly what you did.
You see - we AGREE!

It will be improbable for a given set of numbers to repeat often, and also rare, so as put this way -- we agree.

Yes, we AGREE!

But whether I single out a set of numbers before or after as being specific, it makes no difference as to its likelyhood of occuring (as a specific set).

Yes, the odds are 1 in 13 million, regardless of whether or not you have chosen it. We still AGREE!

...but for any set to occur at all is not improbable.

Yes, one of those 13 million sets of numbers will definitely occur. The probability is 1. Again we AGREE!

...those numbers [1,2,3,4,5,6] have just as much chance of occuring as any other single combination. Is it really so amazing to you -- knowing that?

You are, of course, correct - in a way.
But I am correct as well - in a different way.

I still contend that the majority (a vast majority in my opinion) of bridge players would be amazed if they were dealt 13 hearts. But, if these players think about the odds and realize that every single set of thirteen cards has the same improbable odds as the thirteen hearts, they would wonder whether they should have been amazed. Of course the first reaction is emotional, the second is rational. The emotional reaction is the result of the recognition of a PATTERN in the hand dealt and the immediate recognition of the extremely improbable odds of this pattern. Any other odd pattern looks like any other odd pattern, and therefore the reaction does not occur.
So you see - and this is absolutely marvelous - even though we are saying the exact opposite, we are both correct and we can still both AGREE!

Use colors instead of numbers and all amazement fades away. (Unless they match the ROY-G-BIV pattern, right? ).

Well, now you are stealing from me, so we must definitely be AGREEing with each other here!
And a perfect illustration of the reason why those who are amazed are amazed: PATTERN!
(re ROYGBIV: you will not believe this JT, but I was thinking exactly the same thing, when I first thought of the coloured balls.)

Perhaps it's because of the above I am not amazed so much.

Yes, if you think about it, it is not so amazing but (if you don't think about it) you can't really help but be amazed when that PATTERN comes up!
So, you see, with a little effort, we really can AGREE that we are both correct!


regards,
BillyJoe

 

[Just thinking]

You forgot to read the highlighted bit. When you do , you will see that I said exactly what you did.
You see - we AGREE!

I guess it was the context in which you used the word individually that caused my red flag to go up. (BTW, since we do agree on everything else, we can delete continuing comment on it.)

A single individual set of 6 lotto numbers has a small probability of occuring, yes, but (and this is a big but) we can not look at that probability and attach it to the event of drawing a set of lotto balls. Why? Because there are now two different events being described.

Event 1) Drawing a set of lotto balls (just 6 numbers in general).

Event 2) Drawing a specific individual set.

The confusion seems to be emerging when one describes Event 1 and then considers the odds of getting those specific numbers drawn. But saying that now describes Event 2. Event 1 will result in a rare combination (as will Event 2), but only Event 2 is improbable. Whether one singles out an individual drawing (outcome) after it happens or predicts a specific combination beforehand, it makes no difference -- one event out of millions is being identified, making that result improbable as compared to all the others. It is not fair (or correct) to believe that since that individual result is so improbable, what was just done (drawing 6 numbers -- and then looking at those 6 that turned up) was an improbable event.

I challenge you (or anyone else) to describe and event that is improbable and happens often. It must not be event specific -- meaning in no way can one outcome (or repeat event) be somehow singled out, either before or after the occurance of the event -- unless of course it happens often as compared to the number of repeat trials.

Good luck.

 

[BillyJoe]

A single individual set of 6 lotto numbers has a small probability of occuring, yes, but (and this is a big but) we can not look at that probability and attach it to the event of drawing a set of lotto balls. Why? Because there are now two different events being described.

Event 1) Drawing a set of lotto balls (just 6 numbers in general).

Event 2) Drawing a specific individual set.

The confusion seems to be emerging when one describes Event 1 and then considers the odds of getting those specific numbers drawn. But saying that now describes Event 2. Event 1 will result in a rare combination (as will Event 2), but only Event 2 is improbable.

I am now sufficiently conversant with your idiosyncratic style of writing to be able to see that I do actually agree with you, even though it looks like the exact opposite of what I would have written
(All jokes aside, I do understand what you have just written and I agree.)

I challenge you (or anyone else) to describe and event that is improbable and happens often.

I won't attempt to describe such a scenario, because I have never claimed that an event that is improbable can happen often. To me it is a contradiction in terms.

What I have said is.....
Improbable events happen all the time.
Which I qualified as....
Events, which are members of the class "Improbable Events", happen all the time.
And which, in JT speak, comes out as....
Improbable events (as a group) happen all the time.

I trust we both AGREE with that.

BJ